Abstract
Some Colombian commercial banks have used the strategy of offering promotional prizes in order to attract new savings customers. In this paper we develop a two-stage game model that allows us to understand the effects of this promotional strategy on the deposit interest rates, the deposit market shares and the intermediation spreads. We find that under this strategy it is possible for the bank that offers the highest prize to segment the deposit market serving only customers that assign high subjective probabilities to winning prizes. More importantly we show that the bank that offers the highest promotional prize not only pays the lowest deposit interest rate but also has the largest deposit market share and the widest intermediation spread.
Keywords: Banking Competition, Interest Rates and Interest Rate Spreads. JEL Classifications: D21, D43, G21 and L13.
∗The author thanks Rafael Rob, Manuel Willington and Leslie Stone for their helpful comments. All errors remain mine.
†Department of Economics, University of Pennsylvania, 3718 Locust Walk, Philadelphia, PA19104-6297.
Since the mid 1990’s, some Colombian commercial banks have offered prizes to attract new customers who want to open savings accounts. Cars, houses, and cash have been some of the prizes offered to try to capture the attention of new depositors. The new promotion strategy works like a lottery or a raffle, in the sense that it is not certain that potential new customers will win the prize. Despite the uncertainty of the prize, depositors have welcomed the new banking promotional strategy. This fact suggests that Colombian commercial banks have found a new strategy to compete in the deposit market as an alternative to using interest rates.
It is clear that this innovative activity of promotional prizes must respond to some eco-nomic forces. At first glance the motivation of this activity can be explained by the banks’ desire of acquiring a bigger deposit market share. However, the explanation is not as simple. As suggested by Silber (1983), newfinancial practices are in general designed to lessen the constraints imposed on banks in order to achieve a particular goal. These constraints can be external and internal such as government regulations, high interest rates, high inflation rates and the type of market structure, among others. In this sense, Colombian banks’ strategy of offering prizes can be understood as a way to loosen some of their internal and external constraints in order to increase their deposits or their access to funds sources.
The type of constraints may vary across banks. However there are some constraints that are common for all the banks. For instance, after implementing diversefinancial innovations, banks repeatedly face the internal constraint of having the interest rate as the only simple instrument to stimulate deposits. But offering higher interest rates to attract deposits also implies rising the costs paid by the bank. In addition if market interest rates are already high or the government regulates them, then the use of the interest rates as a tool to capture more savings can be externally constrained.
Some empirical evidence from Colombia supports the importance of the aforementioned external constraints. In particular it has been argued that the interest rates have been high during the last decade. According to Greco (1999) in the period 1990 to 1997, the nominal annual interest rate on loans oscillated roughly between 34% and 47%, while the annual nominal interest rate on deposits oscillated between 23% and 37%. These levels of interest rates were considered high since in the same period there were several episodes in which the Central Bank tried to lower and regulate them.
can be understood not only as a strategy to loosen some contraints in order to stimulate deposits. It can be also understood as an attempt to relax these constraints to increase their intermediation spread up to their historical high levels. In fact, Barajas et al. (1999) mention that the interest rate spread declined steadily from an initial level of about 25% in 1991 to 19% in 1996.
Based on all these stylized facts, in this paper we develop a model that explains how offering prizes to their new savings customers, banks can reduce the interest rate paid on deposits, increase their intermediation spread, and stimulate the demand for deposits.
We model Banks as firms that compete in the deposits market. They supply savings
accounts that can be characterized by two features: first, their interest rate and second,
their potential prize.
More formally we construct a model in which banks differentiate horizontally and verti-cally.1 The horizontal differentiation is based on the different and exogenous physical location of the banks, which affects their interest rate on deposits due to transportation costs. The vertical differentiation comes from the banks’ strategy of offering promotional prizes to their customers. In this sense the prize can be viewed as an improvement in the quality of the savings accounts.
We model competition between banks as a two-stage game duopoly. In the first stage
banks simultaneously decide the value of the prize they offer in order to attract new savings customers. In the second stage, banks compete with the interest rate that they pay on deposits.
It is important to note that we do not explicitly model the decision of offering and not-offering a prize. This decision is implicitly taken by each bank when it decides to offer a positive value of the prize or a prize whose value is zero.2 Moreover, the goal of our model is not to explain why some banks have not adopted the prize strategy. Instead we want to use the model to explain how the aforementioned strategy has affected interest rates spreads and deposits in the banks that adopted the strategy.
The prize strategy can be seen as an instrument that segments the depositors according to the subjective probability that these customers assign to win the prize. This probability is
1Two products are horizontally differentiated when there is no ranking among consumers based on their
willingness-to-pay for the two products. On the other hand two products are vertically differentiated when there exits such a ranking. See Neven and Thisse (1982).
2To model the decision of offering and not-offering a prize we can include another stage at the beginning
a measure of the their taste for raffles and lotteries, and can be considered as the probability-type of each individual.3 Depending on the unitary cost of transportation and the difference between the prizes’ values, there are two types of segmentation or dominance. If the unitary cost of transportation for the savings customers is greater than the difference between the prizes’ values, then the banks serve all the customers’ probability-types, regardless of the subjective probability that the depositors assign to winning the prize. This case is described as horizontal dominance. On the other hand, if the unitary cost of transportation is less than the difference between the prizes’ values, then the bank that offers the highest prize attracts only customers that assign a subjective high probability of winning the prize. In other words, this bank ends with a zero market share for low probability-type depositors. This situation corresponds to vertical dominance.
We find that the equilibrium of the two-stage game depends on the type of dominance.
Under horizontal dominance we obtain a symmetric equilibrium. That is, both banks offer
the same prize, the same deposit interest rates and have the same deposit market share.
Under vertical dominance we derive an asymmetric equilibrium where only one bank offers
a prize. This bank pays the lowest interest rate and has the largest deposit market share and the widest intermediation spread. Moreover this bank specializes completely in serving customers that assign a subjective high probability to winning the prize, while the bank whose prize is zero specializes in serving low probability-type customers.
We also define and solve a benchmark game. This is a simple game in which banks
only compete with deposit interest rates. Comparing the equilibrium of this game with
the equilibrium of the two-stage game lead to some interesting results. We find that under
horizontal dominance, the equilibrium interest rates and the market shares of the two stage
game are the same as the ones that are obtained under the benchmark game. On the other
hand, under vertical dominance, the bank that offers a positive prize reduces its deposit
interest rate and increases its deposit market share in comparison with the ones that it has
under thebenchmark game. Furthermore the bank that does not offer any prize pays a higher
interest rate and has a lower deposit market share than those it has under the benchmark
game.
Our model is based on the large literature of industrial organization that has studied
price competition among firms under horizontal differentiation and/or under vertical
differ-3The word probability-type is introduced since the type of each individual is defined by the location of
entiation. Some examples of models of horizontal differentiation, also known as the “address location” approach, are Hotelling (1929), d’Aspremont et al. (1979), Salop (1979) and Economides (1989a). Some prototypes of models of vertical differentiation are Mussa and Rossen (1978) and Shaked and Sutton (1983). In addition, Economides (1989b), Econo-mides (1993), Neven and Thissen (1990) and Dos Santos and Thisse (1996) are models that simultaneously study and integrate vertical and horizontal differentiation considering solely two characteristics of the products: variety and quality.
There are also articles that have been written to explain the industrial organization of banking including the concepts of vertical or horizontal differentiation. Some of them relied on previous results derived in the price competition papers aforementioned. In general the structure of these banking models is based upon the circular-city model of Salop (1979). Adapting this structure to banking competition has allowed researchers to study different
problems. For instance, Freixas and Rochet (1997) use this structure to find the optimal
number of banks while Chiappori et al. (1995) apply it to study the impact of deposit rate regulation on credit rates. In addition, Matutes and Padilla (1994) utilize this structure to find the appropriate level of interbank cooperation in automated teller machine (ATM) networks, whereas Bouckaert and Degryse (1995) apply it to analyze the consequences of the introduction of phone banking. Most of these models do not consider vertical differentiation among banks. If they consider it, like in Bouckaert and Degryse (1995), they assume that the depositors have the same taste for the quality-option that the banks offer. Therefore it is important to emphasize that the model that we develop in this paper makes explicit the interaction between vertical and horizontal product differentiation as Degryse (1996) does to analyze the conditions under which banks offer remote access.
We assume that there is a duopolistic industry of banks in the deposit market.4 The two banks are denoted by Bank 1 and Bank 2. Each bank has a single branch that is located on a circle with unit circumference. For simplicity we suppose that the banks are already located at a distance of 1/2 from each other.
We model competition for deposits as a two-stage game. In the first stage banks
simul-taneously decide the value of the prize, qi [0, ), that they offer to attract new savings
customers. In the second stage, banks compete with deposit interest rates, ri [0, ).
Following Economides (1993), we argue that this timing is justified by the fact that all strategic variables are not equally flexible. In the short-run, interest rates can be easier to change than the prize strategy. The reason is that there is a promotional and advertising campaign associated with the prize strategy that takes time to design and it may be difficult to change.
From the timing of the game and the fact that banks’ locations are given exogenously it is evident that our purpose in this model is to focus on the introduction of prizes and their interaction with interest rates and deposit market shares. However it is important to notice that although banks do not choose their location, their deposit accounts are still characterized by two features: first, the exogenously given location of the bank that implicitly defines the physical accessibility to it and affects the deposit interest rate through transportation costs; second, the value of the prize.
As in many of the papers of banking, that were mentioned above, we take the existence of banks as given and concentrates only on their liability side.5 The deposits, D
i, that the
banks attract, are invested to obtain an identical and fixed returnR ri per unit of money.
Therefore the intermediation spread can be defined as (R ri).
Each bank i maximizes profits, πi
πi = (R ri)Di C(qi) (1)
4For the purpose of this paper it is sufficient to consider only two banks. An extension to more banks is
straightforward. However graphical analysis becomes cumbersome for models with more than two banks. 5We are excluding the possible competition that banks can have in the credit market. This exclusion
z
θ
Figure 1:
where C(qi) represents the cost function for the bank of offering the prize qi. We assume
that the function C(qi) is increasing at an increasing rate (C0(qi) > 0 and C00(qi) > 0).
The intuition of this assumption is that each bank is willing to offer higher prizes to attract more customers; but this implies that it also has to invest more resources in advertising and promotion to reach not only the individuals near the bank but also the individuals that are closer to its competitor.
We assume that each depositor is endowed with one unit of money that she invests at only one of the two banks. Depositor preferences vary along two dimensions. First, each depositor
has a unique location z on the circumference with z [0,1] and measured with respect to
the bank i. Second, each depositor has a taste for raffles or prizes. This taste is represented by the subjective probability that individuals assign to winning the prize. This probability
is denoted by θ [0,1] and also describes the probability-type of each depositor. In this
sense individuals differ in their probability-type.
It is important to emphasize that it is not certain that when a depositor opens a new savings account in a bank she will win the prize offered by this bank. She only gets the option of participating in a raffle.
Under these assumptions we can characterize each depositor by the type (θ, z). As in
We assume that depositors are risk-neutral. The expected utility for each individual of
depositing one unit of money in the bank i can be expressed as
U(qi;z,θ) =E(u(qi;z,θ)) = θ[ω+ri tz+qi] + (1 θ)[ω+ri tz]
Thus,
U(qi;z,θ) =ω+ri tz+θqi (2)
where ω is the reservation value that we suppose to be large enough such that the deposit
market is covered; andt is the unitary cost of transportation. As Matutes and Vives (1996) point out, it is interesting to observe that the transportation costs, tz, can be understood in different ways. They do not only represent the depositors’ cost of time spent traveling to the bank but also represent the bank’s provision of diverse services such as ATM network sizes or consumer credit facilities, among others.
We proceed to derive the demand for deposits. As it was stated before we assume that
there are only two banks i = 1,2. Without loss of generality we suppose that the Bank
1 offers a higher prize than the Bank 2, q1 q2. Using the utility function in (2), it is
possible to derive the set of depositors that are indifferent between Banks 1 and 2. For any probability-type θ [0,1], the marginal depositor is found by solving for the location z that makes her indifferent between the two banks. In other words, using the expected utility (2) the location z solves6
ω+r1 tz+θq1 =ω+r2 t 1
2 z +θq2
and defining x(θ) = 2z(θ) we obtain
x(θ) = 2z(θ) = 1
t r1 r2 + t
2 +θ(q1 q2) (3)
where x(θ) represents the market share of Bank 1 for the probability-types θ. This is a
linear and increasing function in θ that partitions the total deposit market in two groups of depositors. It defines the market area of each bank, as illustrated in Figure 2. An increase in r1 (decrease in r2), shifts the function to the left increasing the market area of Bank 1 and reducing the market area of Bank 2.
6Note that for each individual the assigned probability of winning the prizeq
Figure 2:
To determine the demand for deposits of Bank 1, D1,as a function of the interest rate
r1,we integrate the function x(θ) of the equation (3) over [0,1] taking into account the
appropriate range of r1. There are 5 ranges and we continue describing them.
Givenr2, for “very low” values ofr1,and regardless of the magnitude of the slope (q1−q2)
t ,
the line x(θ) does not cross the unit square in Figure 2. This defines the segment of the demand that we call DI1. For “low” values of r1, given r2 and regardless of
(q1−q2)
t , the line
x(θ) crosses the bottom and the right side of the unit square in Figure 2. This determines
the segment of the demand that corresponds to DII
1 . Givenr2 and for “intermediate values” r1 the function x(θ) can cross either the vertical sides or the horizontal sides of the unit square. The first possibility implies that
∂x(θ)
∂θ =
(q1 q2)
t <1 (q1 q2)< t
This means that the difference between the value of the prizes offered by the banks (q1 q2) is less than the unitary cost of transportation, t. This situation is known as
horizontal dominance (q1 q2 < t).7 Under this situation, Bank 1 attracts a strictly positive
(13) is violated
Bank 2’s Share
(12) is violated Bank 1’s
Share
Figure 3:
market of the low probability-type depositors, but it does not serve the entire market for the high probability-type depositors (see Figure 3).
In the second possibility
∂x(θ)
∂θ =
(q1 q2)
t >1 (q1 q2)> t
In this situation, the difference between the value of the prizes offered by the banks (q1 q2) is greater than the linear rate cost of transportation, t. This situation is referred to as vertical dominance (q1 q2 > t). Under this situation, Bank 1 attracts the entire market of the high probability-type depositors and has a zero market share for the low probability-type depositors (see Figure 4).
It is important to notice that if
∂x(θ)
∂θ =
(q1 q2)
t = 1 (q1 q2) =t
there is neither vertical nor horizontal dominance.
Under these two situations,vertical dominance and horizontal dominance, the segment
of the demand function for deposits will be called DIII
1 .
For “high” values of r1, given r2 and independently of the slope (q1−q2)
t , the line x(θ)
(23) is violated
(22) is violated Bank 2’s
Share Bank Share 1’s
Figure 4:
demand that we call DIV
1 . Finally, given r2, for “very high” values of r1, and independently of the slope (q1−q2)
t , the linex(θ) does not cross the unit square. This defines the segment of
the demand that corresponds to DV
1 .
The details for the derivation of the demand for deposits for bank 1, D1, can be found
in the Appendix 1. In this part of the paper we only present the derived functional forms for this demand and give specific meaning to the aforementioned terms “very low”, “low”, “intermediate”, “high” and “very high”, through closed intervals for the deposit interest rate r1.
The results (derived functional form) can be summarized as follows. First, for a “very
low” interest rate, r1 [0, r2 2t (q1 q2)] = [0, r1min], and regardless of the type of
dominance, it can be established that
DI1 = 0 (4)
Second, ifhorizontal dominance (q1 q2 < t) prevails, then a “low” interest rate satisfies r1 [r2 2t (q1 q2), r2 2t] = [rmin
1 , r11,h]. On the other hand, if vertical dominance
(q1 q2 > t) prevails, then a ”low” interest rate satisfiesr1 [r2 2t (q1 q2), r2+2t (q1 q2)] = [rmin
That is
D1II =
1
2t(q1 q2) r2 r1
t
2 (q1 q2)
2
(5)
Third, under horizontal dominance (q1 q2 < t), when the “intermediate” interest rate
satisfies r1 [r2 t2, r2+2t (q1 q2)] = [r11,h, r12,h], the demand function for deposits is
D1III,h = 1
t r1 r2+ t
2 +
(q1 q2)
2t (6)
Under vertical dominance (q1 q2 > t), when the “intermediate” interest rate satisfies r1 [r2+2t (q1 q2), r2 2t] = [r1
1,v, r21,v],the demand function for deposits corresponds to
DIII1,v =
1
(q1 q2)[r1 r2+ (q1 q2)] (7)
Fourth, wefind that ifhorizontal dominance (q1 q2 < t) prevails then a “high” interest rate satisfiesr1 [r2+2t (q1 q2), r2+2t] = [r2
1,h, r1max].On the other hand, if vertical dominance (q1 q2 > t) prevails then a “high” interest rate satisfies r1 [r2 t
2, r2+
t
2] = [r 2
1,v, rmax1 ]. However the same functional form applies for both types of dominance. That is
D1IV = 1
r2 r1+ t
2 2
2t(q1 q2) (8)
Finally, for the last range, under both types of dominance and for a “very high” interest
rate r1 [r2+ t
2, ) = [r max
1 , ), we establish that
DV1 = 1 (9)
It is important to observe that under neithervertical dominance norhorizontal dominance
(q1 q2 =t), the demand function for deposits can be described solely by the segments DI
1, DII
1 , DIV1 andDV1.Moreover regardless of the type of dominance, the segmentsD1II, D1III and DIV
1 are strictly convex, linear and strictly concave, respectively. However differentiability is not assured.8 Figure 5 illustrates a typical demand function for the deposits of Bank 1.
The demand function for deposits of Bank 2 with respect tor2 can be derived similarly
as we derived the demand function for Bank 1. The shape is the same as the one in Figure
5 . Moreover note that the relation D2 = 1 D1 holds,since the whole market for deposits
is assumed to be covered.
1min 11 12 1max
Figure 5:
The game of this model can be solved applying the principle of backward induction. This means that wefirst, solve the deposit interest rate subgame (second stage) and then we solve the prize game (first stage).
This subgame is a typical Bertrand competition game applied to the Banking industry. Given t, q1 and q2, there are six types of equilibria in this subgame or second stage: three under
vertical dominance and three under horizontal dominance.9
In other words for each type of dominance there are three cases that determine the type
of equilibrium achieved in this stage:
Case 1: The equilibrium appears on the linear segments of the demand for deposits
of each bank, that is DIII
1 and D2III.
Case 2: The equilibrium occurs on the strictly concave segment of D1 and on the
strictly convex segment of D2. This means that the equilibrium occurs on D1II and
DIV2 .
Case 3: The equilibrium rises on the strictly convex segment ofD1 and on the strictly
concave segment of D2.This means that the equilibrium is present on DIV
1 and DII2 .
Before calculating these equilibria it is important to notice their existence can be proven by the following argument. As is shown in the Appendix 2, the profits functions, for the banks, πII
i , πIIIi and πIVi where i = 1,2, are strictly convex, strictly concave and strictly
concave in ri respectively. This implies that these functions are also strictly quasiconcave
along their respective interval of interest rates. Moreover they are continuous on the interest rates of the banks but not necessarily differentiable. In addition, the intervals of the interest rates over which these profit functions are defined, are non-empty, convex and compact
subsets of the Euclidean Space R. Therefore all the conditions are satisfied to assure the
existence of a Nash Equilibrium in pure strategies in each subgame.10 Furthermore, since
the profit functions are strictly quasiconcave in ri a unique equilibrium is assured for the
respective subgames.
The equilibria of the subgames are presented in this part of the paper. In each case, vertical and horizontal dominance are analyzed sequentially.
Case 1
Underhorizontal dominance (q1 q2 < t) and ifr1 [r2 t
2, r2+
t
2 (q1 q2)] = [r 1
1,h, r21,h]
and r2 [r1 t
2 + (q1 q2), r1+
t
2] = [r 1
2,h, r22,h], then the profits maximization problems for
the banks can be formulated as
M ax
r1
πIII1,h =M ax r1
(R r1)D1III,h C(q1)
M ax
r2
πIII2,h =M ax
r2
(R r2)(1 DIII1,h) C(q2)
The equilibrium for this Bertrand competition is characterized by
r1∗,h=R
t 2
(q1 q2)
6 (10)
r2∗,h=R t
2 +
(q1 q2)
6 (11)
In addition this equilibrium is unique and satisfies thatr∗
1,h [r∗2,h t
2, r∗2,h+ t
2 (q1 q2)] and r∗
2,h [r1∗,h t
2 + (q1 q2), r∗1,h+ t
2] if and only if
3t
2 (q1 q2) (12)
and
3t
4 (q1 q2) (13)
In other words (10) and (11) represent the interest rate equilibrium for the parameter region defined by (12) and (13). See Figure 3. Based on this equilibrium we can calculate the deposits demands, profits and intermediation spreads for both banks. They are
DIII1,h∗ = 1
2 +
(q1 q2)
6t (14)
DIII2,h∗ = 1 2
(q1 q2)
6t (15)
πIII1,h∗ =t
1
2 +
(q1 q2)
6t 2
C(q1) (16)
πIII2,h∗ =t 1 2
(q1 q2)
6t 2
C(q2) (17)
R r∗1,h = t
2 +
(q1 q2)
6 (18)
R r∗2,h = t 2
(q1 q2)
6 (19)
On the other hand, under vertical dominance (q1 q2 > t) and if r1 [r2 + 2t (q1
q2), r2 2t] = [r1
1,v, r12,v] and r2 [r1 + t
2, r1
t
2 + (q1 q2)] = [r
1
2,v, r22,v], then the profits
maximization problems for the banks can be stated as
M ax
r1
πIII1,v =M ax
r1 (R r1)D
III
M ax
r2
πIII2,v =M ax
r2 (R r2)(1 D
III
1,v) C(q2)
The equilibrium for this Bertrand competition corresponds to
r1∗,v =R
2(q1 q2)
3 (20)
r∗2,v =R (q1 q2)
3 (21)
This equilibrium is also unique and satisfies thatr∗
1,v [r2∗,v+2t (q1 q2), r2∗,v 2t] and r∗
2,v [r1∗,v+ t
2, r∗1,v t
2 + (q1 q2)] if and only if
(q1 q2) 3t
4 (22)
and
(q1 q2) 3t
2 (23)
In other words (20) and (21) represent the interest rate equilibrium for the parameter region defined by (22) and (23). See Figure 4. We can use these interest rates to derive the deposits demands, profits and intermediation spreads for both banks. They are
D1III,v∗ = 2
3 (24)
D2III,v∗ = 1
3 (25)
πIII1,v∗ =
4(q1 q2)
9 C(q1) (26)
πIII2,v∗ =
(q1 q2)
9 C(q2) (27)
R r∗1,v = 2(q1 q2)
3 (28)
R r2∗,v= (q1 q2)
Case 2
For this case the equilibrium deposit interest rates have the same functional form under both types of dominance. The only particular feature for each type of dominance is the range or intervals of the interest rates for which this equilibrium is valid. Moreover, the intermediation spreads, the profits functions and the demand functions evaluated at the equilibrium interest rates have the same functional form under both types of dominance.
The profits maximization problems for the banks can be stated as
M ax
r1
πII1 =M ax
r1
(R r1)D1II C(q1)
M ax
r2
πIV2 =M ax
r2
(R r2)(1 DII1 ) C(q2)
Define for simplicity∆q= (q1 q2).The equilibrium for this Bertrand competition is defined by11
r∗∗1 =R
t+ 2∆q+ (t+ 2∆q)2 + 32∆qt
16 (30)
r2∗∗=R 3 (t+ 2∆q)
2
+ 32∆qt
16 +
5(t+ 2∆q)
6 (31)
Using these equilibrium interest rates we can determine the deposits demands, profits and intermediation spreads for both banks. This is accomplished in the Appendix 3. We do not present these results in this part since as we will discuss below, Case 2 lacks of importance.
Under horizontal dominance (q1 q2 < t) the equilibrium described by equations (30) and
(31) are valid if r1 [r2 2t (q1 q2), r2 2t] = [rmin
1 , r11,h] and r2 [r1+ t
2, r1 +
t
2 + (q1 q2)] = [r2
2,h, rmax2 ]. Under vertical dominance (q1 q2 > t) the equilibrium characterized by
equations (30) and (31) hold if r1 [r2 2t (q1 q2), r2+ 2t (q1 q2)] = [rmin
1 , r11,v] and
r2 [r1 2t + (q1 q2), r1+ 2t + (q1 q2)] = [r2
2,v, r2max]. Therefore using (30) and (31) we
can conclude that under horizontal dominance the equilibrium interest rate is unique and
satisfies that r∗∗
1 [r∗∗2
t
2 (q1 q2), r∗∗2
t
2] and r∗∗2 [r∗∗1 +
t
2, r∗∗1 +
t
2 + (q1 q2)] if and only if
(q1 q2) 3t
2 (32)
Note that in this case condition (12) is violated.
On the other hand, under vertical dominance the equilibrium given by (30) and (31) is
unique and satisfies that r1∗∗ [r∗∗2 2t (q1 q2), r2∗∗+ 2t (q1 q2)] and r2∗∗ [r∗∗1 2t + (q1 q2), r∗∗
1 + 2t + (q1 q2)] if and only if
3t
4 (q1 q2) (33)
Note that in this case condition (32) is violated.
Finally it is important to notice that when (32) is satisfied with equality (q1 q2 = 32t) then r∗∗
1 =r1∗,hand r2∗∗ =r2∗,h. On the other hand if (33) is satisfied with equality (q1 q2 =
3t
4) then r∗∗1 = r∗1,v and r∗∗2 = r∗2,v. This means that the equilibrium interest rates vary
continuously as q1, q2, and t change.
Case 3
For this case, the equilibrium deposit interest rates also have the same functional forms under both types of dominance. As in Case 2, the only particular characteristic for each type of dominance is the range or intervals of the interest rates for which this equilibrium holds. Moreover the intermediation spreads, the profits functions and the demand functions evaluated at the equilibrium have the same functional form under both types of dominance.
The profits maximization problems for the banks can be stated as
M ax
r1
πIV1 =M ax
r1
(R r1)D1IV C(q1)
M ax
r2
πII2 =M ax
r2
(R r2)(1 DIV1 ) C(q2)
Define for simplicity ∆q = (q1 q2). The equilibrium for this Bertrand competition is
described by12
r1∗∗∗=R+
5t 3 t2+ 32∆qt
16 (34)
r∗∗∗2 =R
t+ t2+ 32∆qt
16 (35)
Using these equilibrium interest rates we can determine the deposits demands, profits and intermediation spreads for both banks. See Appendix 3. We do not present these results
in this part since as we will argue below, Case 2 lacks of importance. Under horizontal
dominance (q1 q2 < t) the results presented by equations (34) and (35) are valid if r1 [r2 + 2t (q1 q2), r2+ 2t] = [r2
1,h, rmax1 ] and r2 [r1
t
2, r1
t
2 + (q1 q2)] = [r
min 1 , r12,h].
Under vertical dominance (q1 q2 > t) the results presented by equations (34) and (35) hold if r1 [r2 t2, r2 + 2t] = [r2
1,v, rmax1 ] and r2 [r1
t
2, r1+
t
2] = [r min
1 , r21,h]. Therefore using
(34) and (35) we can derive that under horizontal dominance this equilibrium is unique and
satisfies that r∗∗∗
1 [r∗∗∗2 +
t
2 (q1 q2), r∗∗∗2 +
t
2] andr2∗∗∗ [r1∗∗∗
t
2, r1∗∗∗
t
2 + (q1 q2)] if and only if
(q1 q2) 3t
4 (36)
Note that in this case condition (13) is violated. On the other hand, under vertical
dominance the equilibrium given by (34) and (35) is unique and satisfies that r∗∗∗
1 [r2∗∗∗
t
2, r2∗∗∗+ 2t] and r∗∗∗2 [r∗∗∗1 2t, r∗∗∗1 +2t] if and only if 3t
2 (q1 q2) (37)
Note that in this case condition (23) is violated.
Finally it is important to notice that when (36) is satisfied with equality (q1 q2 = 34t) then r∗∗∗
1 =r1∗,hand r2∗∗∗ =r∗2,h. On the other hand if (37) is satisfied with equality (q1 q2 =
3t
2) then r∗∗∗
1 = r1∗,v and r∗∗∗2 = r∗2,v. As was mentioned before this means that the equilibrium
interest rates vary continuously as q1, q2,and t change.
3.1.1 Comparative and Static Analysis
It is interesting to analyze and compare some of the partial results from the second stage of the game. We only analyze the results from Case 1, that is, the results of the interest rate equilibrium that correspond to the linear segments of the demand for deposits of each bank, DIII
1 and D2III. We restrict the analysis to this case, because as will be shown later, for the first stage of the game there are no values of the prizes and the rate of the transportation cost that support the interest rate equilibrium of cases 2 and 3. In this sense these two cases lack of importance.
partial analysis in the sense that we have not yet completed the task of solving the two-stage game. In this sense the following proposition and results should always be understood taking
into account that the prizes q1 and q2 are given in the second stage of the game. However
so far we cannot say anything about the optimality of the given prizes.
Proposition 1 In the second stage of the game, regardless of the type of dominance, vertical
or horizontal, the bank that offered the highest prize in thefirst stage pays the lowest deposit interest rate, acquires the largest deposit market share and enjoys the widest intermediation spread.
Proof. The proof is straightforward. Use (10), (11), (14) and (15) to derive the fol-lowing. If q1 > q2 then r∗
1,h < r∗2,h and r∗1,v < r2∗,v. If q1 > q2 then DIII1,h∗ > DIII2,h∗ and
DIII∗
1,v > D2III,v∗. In addition, since the intermediation margin has the simple form: R ri then the bank that pays the lowest interest rate enjoys also the widest intermediation spread (R r∗
1,h > R r∗2,h and R r1∗,v > R r∗2,v).
Even if Proposition 1 is derived from a partial analysis in the sense that we have not solved the complete two-stage game, it has an important implication. This proposition suggests that under both types of dominance a bank has an incentive to offer higher prizes than its competitor since this strategy assures it a wider intermediation spread and a larger deposit market share.
There are other interesting results that come from this partial analysis. In terms of the equilibrium market share there is a difference between the two kinds of dominance. In qualitative terms, under vertical dominance there is a probability-type segmentation of the market. The bank that offers the highest prize specializes in serving high probability-type depositors while the bank that offers the lowest prize serves low probability-type individuals (see Figure 4). Under horizontal.dominance, both banks serve all the different probability-types individuals. In quantitative terms, note that the market share of the bank that offers
the highest prize is greater under vertical dominance than the one under horizontal
domi-nance. This is due to the fact that under the latter q1 q2 < t, and therefore
D1III,v∗ = 2 3 > D
III∗
1,h =
1
2+
q1 q2 6t
This suggests that the bank that offers better prizes than its competitor also has an incentive to segment the market according to the probability-type of the individuals.
Other results are summarized as follows. Underhorizontal dominance we find that
∂r∗
1,h
∂q1 <0
∂r∗
1,h
∂q2 >0
∂r∗
2,h
∂q2 <0
∂r∗
2,h
∂DIII∗
1,h
∂q1 >0
∂DIII∗
1,h
∂q2 <0
∂DIII∗
2,h
∂q2 >0
∂DIII∗
2,h
∂q1 <0
Under vertical dominance we find that
∂r∗
1,v
∂q1 <0
∂r∗
1,v
∂q2 >0
∂r∗
2,v
∂q2 <0
∂r∗
2,v
∂q1 >0
∂DIII1,v∗
∂q1 = 0
∂DIII1,v∗
∂q2 = 0
∂DIII2,v∗
∂q2 = 0
∂D2III,v∗
∂q1 = 0
It is important to notice that under both types of dominance the equilibrium interest rate paid by a bank decreases as the value of the offered prize by this bank increases. On the other hand, the equilibrium interest rate for a bank increases as the value of the prize offered by
its competitor increases. Furthermore, under horizontal dominance,the demand for deposits
of a bank increases as the value of the prize offered by this bank increases. However, this demand for deposits decreases as the value of the prize offered by its competitor increases.
Under vertical dominance is it striking that there is no effect of the prizes on the demand
for deposits of each bank.
Once the equilibrium of the second stage is found, the first stage of the problem can be
solved to derive the equilibrium prize. In Section 2, we proposed an increasing and convex function C(qi) in qi to describe the costs of the promotional prize strategy. For simplicity
we assume the following functional form for the costs incurred by bank i= 1,2
C(qi) =qi2
It is important to notice that we are implicitly assuming that both banks have the same technology to design a promotion campaign for the prize that they are offering. The only difference in their costs is the prize qi that they are offering.
relevant for their results. We believe the introduction of these costs can affect the results that they obtain.
If we apply the assumption of a costless technology of these papers to the present model, it is necessary to constrain the value of the prizes to an interval [0,q],¯ otherwise the banks can end choosing unrealistically an infinite prize in the first stage. This is the same assumption used in models where the vertical differentiation is related with quality, like in Neven and Thisse (1990), Economides (1989b) and Economides (1993).
Once more, the equilibria for this first stage can be characterized according to the type
of dominance. Therefore we analyze Cases 1, 2 and 3 sequentially.
Case 1
Underhorizontal dominance (q1 q2 < t) the profit maximization problems for the banks can be stated as
M ax
q1
πIII1,h∗ =M ax
q1
t 1
2 +
(q1 q2)
6t 2
q12
M ax
q2
πIII2,h∗ =M ax q2
t 1 2
(q1 q2)
6t 2
q22
and the equilibrium is
q1∗,h=q∗2,h = 1
12 (38)
Using equations (10), (11) and (14)-(19) the equilibrium interest rates, the equilibrium demands for deposits, the equilibrium profits and the equilibrium intermediation spreads are
r∗1,h =r2∗,h=R t
2 (39)
D∗1,h =D∗2,h = 1
2 (40)
π∗1,h =π∗2,h = t 4
1
144 (41)
R r∗1,h =R r∗2,h= t
Under vertical dominance (q1 q2 > t) the profit maximization problems for the banks can be described as
M ax
q1
πIII1,v∗ =M ax
q1 4
9(q1 q2) q
2 1
M ax
q2
πIII2,v∗ =M ax
q2 1
9(q1 q2) q
2 2
and the equilibrium is described by
q1∗,v =
2
9 q
∗
2,v = 0 (43)
Using equations (20), (21) and (24)-(29) the equilibrium interest rates, the equilibrium demands for deposits, the equilibrium profits and the equilibrium intermediation spreads can be expressed as
r1∗,v =R 4
27 r
∗
2,v =R
2
27 (44)
D∗1,v = 2
3 D
∗
2,v =
1
3 (45)
π∗1,v = 4
81 π
∗
2,v =
2
81 (46)
R r∗1,v =
4
27 R r
∗
2,v =
2
27 (47)
These results for both types of dominance are summarized in the following two proposi-tions.
Proposition 2 For the two-stage game, if there is horizontal dominance (q1 q2 < t) and
if the cost functions are described by C(qi) =qi2 where i= 1,2then a symmetric equilibrium is obtained in which both banks select the same prizes and the same deposit interest rates. These in turn imply that both banks acquire the same deposit market shares and receive the same profits.
Proposition 3 For the two-stage game, if there is vertical dominance (q1 q2 > t) and if the
This last proposition is more robust than Proposition 1, since it is a proposition that is derived based on the whole two-stage game. Therefore in contrast to Proposition 1, these
Proposition 2 and 3 suggest that only under vertical dominance a bank has an incentive to
offer higher prizes than its competitor since this strategy assures it a wider intermediation
spread and a larger deposit market share. Moreover notice that under vertical dominance,
Bank 1 segments the deposit market serving only high probability-type customers while Bank 2 specializes in serving low probability-type customers (see Figure 4).
We proceed analyzing Cases 2 and 3. In both cases we argue that there is not a pair of prizes (q∗
1, q2∗) that supports the equilibrium interest rates derived for the second stage
Case 2
The following propositions are useful to discard an equilibrium for this case.
Proposition 4 For the parameter region defined by (q1 q2) 32t and under horizontal
dominance, (q1 q2) < t, there is not a pair of prizes (q∗
1, q∗2) that supports the equilibrium
of the interest rate defined by (30) and (31).
Proof. The proof is straightforward. Take the inequalities 2(q1 q2) 3tand 2(q1 q2)<
2t and note that there are no positive values of (q1 q2) that satisfy both inequalities at
the same time. Therefore it is not possible to support the equilibrium of the interest rate defined by (30) and (31).
Proposition 5 For the parameter region defined by 34t (q1 q2) and under vertical
domi-nance, (q1 q2)> t, there is no a pair of prizes (q∗
1, q2∗) that supports the equilibrium of the
interest rate defined by (30) and (31).
Proof. The proof is straightforward. Take the inequalities3t 4(q1 q2)and 3(q1 q2)> 3t and note that there are no positive values of (q1 q2) that satisfy both inequalities at the same time. Hence it is not possible to support the equilibrium of the interest rate defined by (30) and (31).
Case 3
on the prizes. Therefore they cannot be solved analytically.13 However it is possible to solve numerically for the equilibrium prizes. Simulations for different values of the unitary cost of transportation, t, suggest that the equilibrium is asymmetric. We obtain that q1∗ > 0 and q∗
2 = 0, regardless of the type of dominance. However for all the simulations we
find that (q∗
1,q2∗) does not satisfy (36) under horizontal dominance, and (37) under vertical
dominance. Thus we argue that under this case it is not possible to find equilibrium prizes in the first stage that support the equilibrium interest rates described by (34) and (35).
The previous analysis is useful to emphasize that the only equilibria that are relevant for the two-stage game are those derived in the Case 1. Hence we will continue focusing on this case.
In order to understand the impact of the introduction of “prizes” in the banking industry upon interest rates, demand deposits and intermediation spreads, it is useful to pursue the following exercise. We will compare the equilibria of two different duopolistic banking industries. The first industry is characterized by two banks that compete using solely their deposits interest rates. Banks do not offer prizes or raffles. This is defined as thebenchmark
game The second industry is characterized by the two banks that are involved in the two-stage game that we solved above.
We proceed analyzing the benchmark game. In this game, two banks are involved in a
Bertrand competition using as strategies their deposits interest rates. This game is the same game as in Salop (1979) that is applied for the case of banking in Freixas and Rochet (1997).
As before we can find the marginal depositor by solving for the location z that makes
her indifferent between the two banks. In other words using the utilities derived from the services of each bank, the location z solves
ω+r1 tz=ω+r2 t 1
2 z
and defining x= 2z we obtain
x= 2z = 1
t r1 r2+ t
2 (48)
where x represents the market share of Bank 1. Note that in comparison to equation (3),x
in (48) is an independent function of the probability-type of the individuals (θ).
The banks’ problem can be stated as follows. For banki=j
M ax
ri πi =M axri (R ri)Di =M axri (R ri)
1
t ri rj + t 2
Solving this game we find the following Nash Equilibrium
r1B =r2B =R t
2 (49)
and using it we can derive the demands for deposits, the profits and the intermediation spreads for both banks. They are
D1B =D
B
2 = 1
2 (50)
πB1 =π
B
2 = t
4 (51)
R r1B =R r
B
2 = t
2 (52)
This is a symmetric equilibrium in which the market of deposits is divided in equal shares for each bank. The division of the deposits market is represented in Figure 6.
Some interesting results arise comparing the equilibria and outcomes of the benchmark
game and those of the two-stage game (Case 1). The main results of this comparison are summarized in the following propositions.
Proposition 6 If there is horizontal dominance in the described two-stage game then the
deposit interest rates, the deposit market shares and the intermediation spreads for both banks are the same as those of the benchmark game.
Proof. Compare (39), (40) and (42) with (49),(50) and (52).
Proposition 7 If there is vertical dominance in the described two-stage game then the bank
that offers a prize, pays a lower deposit interest rate, has a larger market share and enjoys a wider intermediation spread than those that it has under the benchmark game.
Figure 6:
The last proposition supports the argument by Silver (1983). In particular, notice that under vertical dominance the strategy of offering prizes loosens some of the internal and external constraints for a bank. The new strategy not only endows a bank with a new tool to compete in the deposit market. It also allows the bank to achieve a larger deposit market share and a wider intermediation spread if it offers the biggest prize.
It is relevant to point out that under vertical dominance the profits of the bank that
We construct a model that allows us to understand the motivation of some Colombian banks of offering some promotional prizes and raffles to new potential savings customers. The new strategy endows banks with a new tool to compete in the deposits markets without relying exclusively on the use of deposit interest rates.
We study the impact of this promotional strategy on the deposit interest rates, the deposit
market shares and the intermediation spreads. We find that this impact can be analyzed
by the equilibria of a two-stage game duopoly. In the first stage banks compete with prizes while in the second stage they compete with interest rates.
The types of equilibria that we find depend on the type of dominance that prevails.
There are two types of dominance: horizontal and vertical. Under horizontal dominance
the difference between the value of the prizes offered by the banks is less than the unitary cost of transportation of the customers. The equilibria under this type of dominance imply that both banks serve all the customers regardless of the subjective probability that these
customers assign to winning prizes. Under vertical dominance the difference between the
value of the prizes offered by the banks is greater than the unitary cost of transportation of the customers. In equilibrium this type of dominance allows one bank to segment the deposit market in terms of the customers probability-type. In general, the bank that offers the highest prize also serves only customers that assign high subjective probabilities to winning prizes. On the other hand the bank that offers the lowest prize, or no prize, specializes in serving customers that assign very low probability to win prizes.
From the equilibria of the two-stage game, we derive the following interesting results.
Under horizontal dominance a symmetric equilibrium is obtained in which both banks not
only offer the same prizes’ value but also pay the same deposit interest rates, have the same
deposit market shares and receive the same profits. Under vertical dominance only one bank
offers a positive prize allowing it to segment the deposit market in terms of the customers probability-type. This bank not only pays the lowest deposit interest rate but also has the largest market share and enjoys the widest intermediation spread.
Finally we introduce a benchmark game to be able to understand what has changed in
the Colombian banking structure with the introduction of promotional prize strategies. This
benchmark game is defined as the situation in which banks do not offer any prize and compete only in deposit interest rates. Comparing the equilibrium of this game with equilibrium of the
the deposit interest rates, the deposit market shares and the intermediation spreads are the
same as the ones of the benchmark game. Under vertical dominance, the bank that offers
the highest prize, pays a lower deposit interest rate, has a larger deposit market share and
enjoys a wider intermediation spread than those that it has under the benchmark game.
In this appendix we derive the demand for deposits for Bank 1. To accomplish this task it is necessary to integrate the function x(θ) of equation (3) over [0,1] taking into account the appropriate range of the interest rate r1. There are 5 cases for each type of dominance.
Suppose that r2 is given. For ”very low” values of r1, and independently of the type
of dominance, the line x(θ) does not cross the unit square. This defines the segment of the
demand that we call D1I. Equivalently when r1 [0, r2 2t (q1 q2)] = [0, r
min
1 ] we can
establish that
DI1 = 0 (53)
For ”low” values of r1, given r2 and independently of the slope (q1−q2)
t , the line x(θ)
crosses the bottom and the right side of the unit square. This determines the segment of the demand that we call DII1 . In this case the same demand functional form DII1 holds for both types of dominance. However, the range of the interest rate in which this functional form is valid varies according to the type of dominance. If horizontal dominance (q1 q2 < t)
prevails then r1 [r2 2t (q1 q2), r2 2t] = [rmin
1 , r11,h]. On the other hand if vertical dominance (q1 q2 > t) prevails thenr1 [r2 2t (q1 q2), r2+2t (q1 q2)] = [rmin
1 , r11,v].
In both cases
DII1 = 1
¯
θ
x(θ)dθ= 1
¯
θ
r1 r2+ t
2+θ(q1 q2) dθ
where
¯
θ = 1
(q1 q2) r2 r1
t 2
Then
D1II =
1
2t(q1 q2) r2 r1
t
2 (q1 q2)
2
Givenr2 and for ”intermediate values” r1 the function x(θ) can cross either the vertical sides or the horizontal sides of the unit square. The first case implieshorizontal dominance
and the second case implies vertical dominance. Under horizontal dominance (q1 q2 < t)
we have thatr1 [r2 2t, r2+2t (q1 q2)] = [r1
1,h, r12,h] and the demand function for deposits
can be calculated as
D1III,h = 1
0
x(θ)dθ = 1
0
r1 r2+ t
2+θ(q1 q2) dθ
Then
D1III,h = 1
t r1 r2+ t
2 +
(q1 q2)
2t (55)
On the other hand, if vertical dominance (q1 q2 > t) prevails then r1 [r2+ t
2 (q1 q2), r2 2t] = [r11,v, r12,v] and the demand function can be derived as
D1III,v = ˆθ
¯
θ
x(θ)dθ = ˆ
θ
¯
θ
r1 r2+ t
2+θ(q1 q2) dθ
where
¯θ = 1
(q1 q2) r2 r1
t 2
ˆθ = 1
(q1 q2) r2 r1+
t 2
Then
DIII1,v = 1
(q1 q2)[r1 r2+ (q1 q2)] (56)
For ”high” values of r1, given r2 and independently of the slope (q1−q2)
t , the line x(θ)
crosses the top and the left side of the unit square. This defines the segment of the demand that we call DIV
1 . In this case the same demand functional form D1IV applies to both
types of dominance. However, the range of the interest rate in which this demand holds
varies with the type of dominance. If horizontal dominance (q1 q2 < t) prevails then
r1 [r2 + 2t (q1 q2), r2+ 2t] = [r21,h, rmax1 ]. If vertical dominance (q1 q2 > t) prevails then r1 [r2 2t, r2+2t] = [r21,v, rmax1 ].In both cases
D1IV = ˆ
θ
0
x(θ)dθ+ 1
ˆ
θ
dθ= ˆ
θ
0
r1 r2+ t
2 +θ(q1 q2) dθ+
1
ˆ
θ
where
ˆ
θ= 1
(q1 q2) r2 r1+
t 2
Then
D1IV = 1 r2 r1+
t
2 2
2t(q1 q2) (57)
This determines the segment of the demand that we callDIV
1 .Finally, givenr2, for ”very high” values of r1,and regardless of the type of dominance, the linex(θ) does not cross the
unit square. This defines the segment of the demand that we call DV
1 . Equivalently when
r1 [r2+ t2, ) = [rmax1 , ), we can establish that
DV1 = 1 (58)
Note thatDII1 is strictly convex, D1III,h and D III
1,v are linear, and DIV1 is strictly concave in r1. All of these segments are increasing in r1. To prove it, assume that q1 > q2 and define
∆q=q1 q2, then we have that
∂DII
1
∂r1 = 1
t∆q r1 r min 1 >0
∂2DII
1
∂r2 1
= 1
t∆q >0
∂DIII
1,h
∂r1 = 1
t >0 and independent of r1
∂DIII
1,v
∂r1 = 1
∆q >0 and independent of r1
∂DIV
1
∂r1 = 1 t∆q [r
max
1 r1]>0
∂2DIV
1
∂r2 1
= 1
t(q1 q2) <0
In addition notice that under horizontal dominance and vertical dominance the demand
functions are continuous. That is
D1II(r1 = r11,h) = ∆q
2t =D
III
1,h(r1 =r
1 1,h)
D1II(r1 = r11,v) = t
2∆q =D
III
1,v(r1 =r
1 1,v)
D1III,h(r1 = r21,h) = 1
∆q
2t =D
IV
1 (r1 =r12,h)
D1III,v(r1 = r21,v) = 1 t
2∆q =D
IV
1 (r1 =r 2 1,v)
D1IV(r1 =r1max) = 1 =DV1
It is straightforward to prove the continuity of the profit functions since the demand functions are continuous. Moreover the convexity of the profit functions can be stated as follows.
Underhorizontal dominance,defining∆q=q1 q2 and using the results from Appendix
1 we can deduce that
∂2πII
1
∂r2 1
= 2∂D
II
1
∂r1 + [R r1]
∂2DII
1
∂r2 1
= 1
t∆q r1 r min 1 +
[R r1]
t∆q >0
∂2πIII
1,h
∂r2 1
= 2∂D
III
1,h
∂r1 <0
∂2πIV
1
∂r2 1
= 2∂D
IV
1
∂r1 + [R r1]
∂2DIV
1
∂r2 1
<0
Under vertical dominance, defining ∆q=q1 q2 and using results from Appendix 1 we
can deduce that
∂2πII
1
∂r2 1
= 2∂D
II
1
∂r1 + [R r1]
∂2DII
1
∂r2 1
= 1
t∆q r1 r min 1 +
[R r1]
t∆q >0
∂2πIII
1,v
∂r2 1
= 2∂D
III
1,v
∂r1 <0
∂2πIV1
∂r2 1
= 2∂D
IV
1
∂r1 + [R r1]
∂2D1IV
∂r2 1
<0
ThereforeπII
Case 2
As was stated above under horizontal dominance the results presented by equations
(30)and (31) are valid if r1 [rmin
1 , r11,h] and r2 [r22,h, rmax2 ]. Under vertical dominance the results presented by equations (30)and (31) hold if r1 [rmin
1 , r11,v] and r2 [r22,v, r2max]. The profits maximization problem for the banks can be formulated as
M ax
r1
πII1 =M ax
r1
(R r1)D1II C(q1)
M ax
r2
πIV2 =M ax
r2
(R r2)(1 DII1 ) C(q2)
Define for simplicity∆q= (q1 q2). The FOC’s for this problem can be expressed as
r2 r1 t
2 ∆q
2
+ 2(R r1) r2 r1 t
2 ∆q = 0
2t∆q r2 r1 t
2 ∆q
2
+ 2(R r2) r2 r1 t
2 ∆q = 0
and the solution is
r∗∗1 =R t+ 2∆q+ (t+ 2∆q)
2
+ 32∆qt
16 (59)
r2∗∗=R 3 (t+ 2∆q)
2
+ 32∆qt
16 +
5(t+ 2∆q)
6 (60)
Using (59) and (60) the demand functions for deposits and the profits functions can be written as
DII1 ∗∗= t 2µ2
2t∆q
D2IV∗∗ = 1 t 2µ2
2t∆q
πII1 ∗∗= t3µ3
πIV2 ∗∗= 3tµ 2
t
2 ∆q 1
t2µ2
2t∆q C(q2)
where
tµ= t+ 2∆q+ (t+ 2∆q)
2
+ 32∆qt
8
while the intermediation spreads are
R r1∗∗= t+ 2∆q+ (t+ 2∆q)
2
+ 32∆qt 16
R r∗∗2 = 3 (t+ 2∆q)
2
+ 32∆qt
16
5(t+ 2∆q)
6
Case 3
Under horizontal dominance the results presented by equations (34) and (35) are valid
if r1 [r2
1,h, r1max] and r2 [r1min, r12,h]. Under vertical dominance the results presented by
equations (34) and (35) hold if r1 [r2
1,v, r1max] and r2 [r1min, r12,h]. The profit maximization
problem for the banks can be stated as
M ax
r1
πIV1 =M ax
r1
(R r1)D1IV C(q1)
M ax
r2
πII2 =M ax
r2
(R r2)(1 DIV1 ) C(q2)
Define∆q= (q1 q2). The FOC’s for this problem can be expressed as
r2 r1+ t 2
2
2t∆q+ 2(R r1) r2 r1+ t
2 = 0
r2 r1 + t 2
2
2(R r2) r2 r1+ t
2 = 0
and the solution is
r1∗∗∗=R+
5t 3 t2+ 32∆qt
r∗∗∗2 =R t+ t
2+ 32∆qt
16 (62)
Using (61) and (62) the demand functions for deposits and the profits functions can be written as
D1IV∗∗∗ = 1 t 2γ2
2t∆q
D2II∗∗∗ = t 2γ2
2t∆q
πIV1 ∗∗∗ = 3tγ 2
t
2 1
t2γ2
2t∆q C(q1)
πII2 ∗∗∗ = t 3γ3
4t∆q C(q2)
where
tγ = t+ t
2+ 32∆qt
8
while the intermediation spreads are
R r1∗∗∗ =
5t 3 t2+ 32∆qt
16
R r2∗∗∗ = t+ t
2+ 32∆qt
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